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Fast Algorithms for Computing Continuous Piecewise Affine Lyapunov Functions

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Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 873))

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Abstract

Algorithms that parameterize continuous and piecewise affine Lyapunov functions for nonlinear systems, both in continuous and discrete time, have been proposed in numerous publications. These algorithms generate constraints that are linear in the values of a function at all vertices of a simplicial complex. If these constraints are fulfilled for certain values at the vertices, then they can be interpolated on the simplices to deliver a function that is a Lyapunov function for the system used for their generation. There are two different approaches to find values that fulfill the constraints. First, one can use optimization to compute appropriate values that fulfill the constraints. These algorithms were originally designed for continuous-time systems and their adaptation to discrete-time systems and control systems poses some challenges in designing and implementing efficient algorithms and data structures for simplicial complexes. Second, one can use results from converse theorems in the Lyapunov stability theory to generate good candidates for suitable values and then verify the constraints for these values. In this paper we study several efficient data structures and algorithms for these computations and discuss their implementations in C++.

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References

  1. Ambrosino, R., Garone, E.: Robust stability of linear uncertain systems through piecewise quadratic Lyapunov functions defined over conical partitions. In: Proceedings of the 51st IEEE Conference on Decision and Control, Maui (HI), USA, pp. 2872–2877, December 2012

    Google Scholar 

  2. Baier, R., Grüne, L., Hafstein, S.: Linear programming based Lyapunov function computation for differential inclusions. Discrete Contin. Dyn. Syst. Ser. B 17(1), 33–56 (2012)

    Article  MathSciNet  Google Scholar 

  3. Baier, R., Hafstein, S.: Numerical computation of Control Lyapunov Functions in the sense of generalized gradients. In: Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS), Groningen, The Netherlands, pp. 1173–1180, no. 0232 (2014)

    Google Scholar 

  4. Björnsson, J., Giesl, P., Hafstein, S., Kellett, C., Li, H.: Computation of continuous and piecewise affine Lyapunov functions by numerical approximations of the Massera construction. In: Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles (CA), USA, pp. 55056–5511 (2014)

    Google Scholar 

  5. Björnsson, J., Giesl, P., Hafstein, S., Kellett, C., Li, H.: Computation of Lyapunov functions for systems with multiple attractors. Discrete Contin. Dyn. Syst. Ser. A 35(9), 4019–4039 (2015)

    Article  MathSciNet  Google Scholar 

  6. Björnsson, J., Gudmundsson, S., Hafstein, S.: Class library in C++ to compute Lyapunov functions for nonlinear systems. In: Proceedings of MICNON, 1st Conference on Modelling, Identification and Control of Nonlinear Systems, no. 0155, pp. 788–793 (2015)

    Google Scholar 

  7. Björnsson, J., Hafstein, S.: Efficient Lyapunov function computation for systems with multiple exponentially stable equilibria. Procedia Comput. Sci. 108, 655–664 (2017)

    Article  Google Scholar 

  8. Branicky, M.: Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. 43(4), 475–482 (1998)

    MathSciNet  MATH  Google Scholar 

  9. Clarke, F.: Optimization and Nonsmooth Analysis. Classics in Applied Mathematics. SIAM, Philadelphia (1990)

    Book  Google Scholar 

  10. Doban, A.: Stability domains computation and stabilization of nonlinear systems: implications for biological systems. Ph.D. thesis, Eindhoven University of Technology (2016)

    Google Scholar 

  11. Doban, A., Lazar, M.: Computation of Lyapunov functions for nonlinear differential equations via a Yoshizawa-type construction. IFAC-PapersOnLine 49(18), 29–34 (2016). 10th IFAC Symposium on Nonlinear Control Systems NOLCOS 2016

    Article  Google Scholar 

  12. Giesl, P., Hafstein, S.: Computation and verification of Lyapunov functions. SIAM J. Appl. Dyn. Syst. 14(4), 1663–1698 (2015)

    Article  MathSciNet  Google Scholar 

  13. Giesl, P., Hafstein, S.: Computation of Lyapunov functions for nonlinear discrete systems by linear programming. J. Differ. Equ. Appl. 20, 610–640 (2014)

    Article  MathSciNet  Google Scholar 

  14. Giesl, P., Hafstein, S.: Implementation of a fan-like triangulation for the CPA method to compute Lyapunov functions. In: Proceedings of the 2014 American Control Conference, Portland (OR), USA, pp. 2989–2994, no. 0202 (2014)

    Google Scholar 

  15. Giesl, P., Hafstein, S.: Revised CPA method to compute Lyapunov functions for nonlinear systems. J. Math. Anal. Appl. 410, 292–306 (2014)

    Article  MathSciNet  Google Scholar 

  16. Hafstein, S.: An algorithm for constructing Lyapunov functions, volume 8 of Monograph. Electron. J. Diff. Eqns. (2007)

    Google Scholar 

  17. Hafstein, S.: Implementation of simplicial complexes for CPA functions in C++11 using the armadillo linear algebra library. In: Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH), Reykjavik, Iceland, pp. 49–57 (2013)

    Google Scholar 

  18. Hafstein, S.: Efficient algorithms for simplicial complexes used in the computation of Lyapunov functions for nonlinear systems. In: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH), Madrid, Spain, pp. 398–409 (2017)

    Google Scholar 

  19. Hafstein, S., Kellett, C., Li, H.: Computation of Lyapunov functions for discrete-time systems using the Yoshizawa construction. In: Proceedings of the 53rd IEEE Conference on Decision and Control (2014)

    Google Scholar 

  20. Hafstein, S., Kellett, C., Li, H.: Continuous and piecewise affine Lyapunov functions using the Yoshizawa construction. In: Proceedings of the 2014 American Control Conference, Portland (OR), USA, pp. 548–553, no. 0170 (2014)

    Google Scholar 

  21. Hafstein, S., Kellett, C., Li, H.: Computing continuous and piecewise affine Lyapunov functions for nonlinear systems. J. Comput. Dyn. 2(2), 227–246 (2015)

    Article  MathSciNet  Google Scholar 

  22. Johansson, M.: Piecewise linear control systems. Ph.D. thesis, Lund University, Sweden (1999)

    Google Scholar 

  23. Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Automat. Control 43(4), 555–559 (1998)

    Article  MathSciNet  Google Scholar 

  24. Julian, P.: A high level canonical piecewise linear representation: theory and applications. Ph.D. thesis, Universidad Nacional del Sur, Bahia Blanca, Argentina (1999)

    Google Scholar 

  25. Julian, P., Guivant, J., Desages, A.: A parametrization of piecewise linear Lyapunov functions via linear programming. Int. J. Control 72(7–8), 702–715 (1999)

    Article  MathSciNet  Google Scholar 

  26. Kellett, C.: Converse theorems in Lyapunov’s second method. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2333–2360 (2015)

    Article  MathSciNet  Google Scholar 

  27. Khalil, H.: Nonlinear Systems, 3rd edn. Pearson, London (2002)

    MATH  Google Scholar 

  28. Lazar, M.: On infinity norms as Lyapunov functions: alternative necessary and sufficient conditions. In: Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, USA, pp. 5936–5942, December 2010

    Google Scholar 

  29. Lazar, M., Doban, A.: On infinity norms as Lyapunov functions for continuous-time dynamical systems. In: Proceedings of the 50th IEEE Conference on Decision and Control, Orlando (Florida), USA, pp. 7567–7572 (2011)

    Google Scholar 

  30. Lazar, M., Doban, A., Athanasopoulos, N.: On stability analysis of discrete-time homogeneous dynamics. In: Proceedings of the 17th International Conference on Systems Theory, Control and Computing, Sinaia, Romania, pp. 297–305, October 2013

    Google Scholar 

  31. Lazar, M., Jokić, A.: On infinity norms as Lyapunov functions for piecewise affine systems. In: Proceedings of the Hybrid Systems: Computation and Control Conference, Stockholm, Sweden, pp. 131–141, April 2010

    Google Scholar 

  32. Li, H., Hafstein, S., Kellett, C.: Computation of continuous and piecewise affine Lyapunov functions for discrete-time systems. J. Differ. Equ. Appl. 21(6), 486–511 (2015)

    Article  MathSciNet  Google Scholar 

  33. Marinósson, S.: Lyapunov function construction for ordinary differential equations with linear programming. Dyn. Syst. Int. J. 17, 137–150 (2002)

    Article  MathSciNet  Google Scholar 

  34. Marinósson, S.: Stability analysis of nonlinear systems with linear programming: a Lyapunov Functions based approach. Ph.D. thesis, Gerhard-Mercator-University Duisburg, Duisburg, Germany (2002)

    Google Scholar 

  35. Massera, J.: Contributions to stability theory. Ann. Math. 64, 182–206 (1956). Erratum. Ann. Math. 68, 202 (1958)

    Article  MathSciNet  Google Scholar 

  36. Ohta, Y.: On the construction of piecewise linear Lyapunov functions. In: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 3, pp. 2173–2178, December 2001

    Google Scholar 

  37. Ohta, Y., Tsuji, M.: A generalization of piecewise linear Lyapunov functions. In: Proceedings of the 42nd IEEE Conference on Decision and Control, vol. 5, pp. 5091–5096, December 2003

    Google Scholar 

  38. Polanski, A.: Lyapunov functions construction by linear programming. IEEE Trans. Automat. Control 42, 1113–1116 (1997)

    Article  MathSciNet  Google Scholar 

  39. Polanski, A.: On absolute stability analysis by polyhedral Lyapunov functions. Automatica 36, 573–578 (2000)

    Article  MathSciNet  Google Scholar 

  40. Sanderson, C.: Armadillo: an open source C++ linear algebra library for fast prototyping and computationally intensive experiments. Technical report, NICTA (2010)

    Google Scholar 

  41. Sauer, T.: Numerical Analysis, 2nd edn. Pearson, London (2012)

    MATH  Google Scholar 

  42. Stroustrup, B.: Software development for infrastructure. Computer 45(1), 47–58 (2012)

    Article  Google Scholar 

  43. Vidyasagar, M.: Nonlinear System Analysis. Classics in Applied Mathematics, 2nd edn. SIAM, Philadelphia (2002)

    Book  Google Scholar 

  44. Yfoulis, C., Shorten, R.: A numerical technique for the stability analysis of linear switched systems. Int. J. Control 77(11), 1019–1039 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The author’s research is supported by the Icelandic Research Fund (Rannís) ‘Complete Lyapunov functions: Efficient numerical computation’ (163074-052) and ‘Lyapunov Methods and Stochastic Stability’ (152429-051), which is gratefully acknowledged.

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Authors and Affiliations

  1. University of Iceland, Dunhagi 5, 107, Reykjavik, Iceland

    Sigurdur Hafstein

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  1. Sigurdur Hafstein
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Correspondence to Sigurdur Hafstein .

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  1. King Abdullah II School of Information Technology, The University of Jordan, Amman, Jordan

    Mohammad S. Obaidat

  2. School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON, Canada

    Tuncer Ören

  3. Department of Informatics, Modeling, Electronics and System Engineering, University of Calabria, Rende, Italy

    Floriano De Rango

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Hafstein, S. (2019). Fast Algorithms for Computing Continuous Piecewise Affine Lyapunov Functions. In: Obaidat, M., Ören, T., Rango, F. (eds) Simulation and Modeling Methodologies, Technologies and Applications . SIMULTECH 2017. Advances in Intelligent Systems and Computing, vol 873. Springer, Cham. https://doi.org/10.1007/978-3-030-01470-4_15

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